Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24);(2,4)(5,6,8,7)(10,14)(12,15)(18,19)(20,24);(1,2,3,4)(5,7,8,6)(17,18,21,19)(20,22,24,23);(9,11)(10,15)(12,14)(13,16)(17,19,21,18)(20,23,24,22);(9,12,13,15)(10,11,14,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72297960, #72297978, #72297979 |
| maximal normal subgroup of | #72297960, #72297978, #72297979 |
| maximal quotient of | #72298307 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 463386);